Since the irreducible components of \(D\)  have an intersection matrix that is negative definite, the map \(H_2(D; \mathbb {Q})\rightarrow H_2(M^{\prime }; \mathbb {Q})\)  is injective
and has image a negative-definite subspace of dimension \(\nu (p-1)\)  (there are \(\nu \)  pairwise disjoint copies of chains of length \(p-1\) .) Its orthogonal complement is the image of \(H_2(M^{\prime }\setminus D^{\prime }; \mathbb {Q})\) .
The transfer map identifies
\(H_2(M^{\prime }\setminus D; \mathbb {Q})\)  with \(H_2(M\setminus M^G; \mathbb {Q})^G\)  and multiplies the intersection pairing with \(p\) . The latter has signature \((3, 19-(p-1)\nu )\) . Hence
\(H_2(M^{\prime })\)  has signature \((3, 19-(p-1)\nu + (p-1)\nu )=(3,19)\) . Since \(M\)  has a unique spin structure, \(G\)  must leave that structure invariant and so \(M^{\prime }\)  inherits
a spin structure. Wu's formula then implies that the intersection form on \(M^{\prime }\)  is even.
Freedman's classification of closed, simply-connected 4-manifolds (Theorem 1.5 of [1]})
then implies that \(M^{\prime }\)  must be homeomorphic to a
\(K3\)  manifold.
